nLab
ordinary mathematics

In the foundations of mathematics, one sometimes comes across the phrase ‘ordinary mathematics’. This seems to mean mathematics based on foundations that are no stronger than necessary to do the standard mathematics of the 20th century.

One might define it (if a formal definition is desired) as mathematics that can be formalised in ETCS; Wikipedia defines it as mathematics that takes place in the von Neumann universe of rank ω+ω\omega + \omega (which is a model of ETCSETCS). One can also think of this as allowing only finitely many applications of the power set axiom (since replacement is needed to iterate this transfinitely) starting with the set of natural numbers.

More generally, in the context of any “unusual” sort of mathematics (not limited to using high-powered foundational axioms), the term “ordinary mathematics” might be used informally to refer to mathematics without that particular unusual aspect.